doc-en/par-transform-en.tex

   1 \section{The option \texttt{transform}}
   2
   3 The option \Lkeyword{transform}, which is nothing else than a formula $\mathbb{R}^3 \rightarrow  \mathbb{R}^3$,
   4 which is applied to every point of the solid. In the first example, the object that accepts the transformation is a cube.
   5 The referenced cube is yellow, the transformed cube is green and the cube before the \Index{transformation} is setup with a reticule.
   6
   7 \subsection{Identical scaling factor in the three coordinates}
   8
   9 The scaling factor is set to $0.5$. It is either introduced within the PostScript variable `\texttt{/Facteur}':
  10 \begin{verbatim}
  11 \pstVerb{/Facteur {.5 mulv3d} def}%
  12 \end{verbatim}
  13 and then passed to the option \verb+transform+:
  14 \begin{verbatim}
  15 \psSolid[object=cube,a=2,ngrid=3,
  16          transform=Facteur](2,0,1)%
  17 \end{verbatim}
  18 or directly passed to the option:
  19 \begin{verbatim}
  20 \psSolid[object=cube,a=2,ngrid=3,
  21          transform={.5 mulv3d}](2,0,1)%
  22 \end{verbatim}
  23 Here the \textit{jps} abbreviation \texttt{transform=\{.5 mulv3d\}} for a function $\mathbb{R}^3 \rightarrow \mathbb{R}^3$ was used.
  24
  25 Another method would be to use the code
  26 \begin{verbatim}
  27 \defFunction[algebraic]{matransformation}(x,y,z)
  28    {.5*x}
  29    {.5*y}
  30    {.5*z}
  31 \end{verbatim}
  32 and then pass it to the option
  33 \texttt{transform=matransformation}.
  34 \begin{LTXexample}[pos=t]
  35 \psset{viewpoint=20 60 20 rtp2xyz,lightsrc=viewpoint,Decran=20}
  36 \begin{pspicture}(-5,-3)(6,5)
  37 \psSolid[object=grille,base=-4 4 -4 4,fillcolor=red!50]%
  38 \axesIIID(0,0,0)(4,4,4)%
  39 \psSolid[object=cube,fillcolor=yellow!50,
  40       a=2,ngrid=3](-2,0,1)
  41 \psSolid[object=cube,fillcolor=green!50,
  42       a=2,transform={.5 mulv3d},
  43       ngrid=3](2,0,1)
  44 \psSolid[object=cube,
  45       action=draw,
  46       a=2,ngrid=3](2,0,1)
  47 \end{pspicture}
  48 \end{LTXexample}
  49
  50 \encadre{The scaling factor also affects the position coordinates of the cube's center.}
  51
  52 \subsection{Different scaling factors for the three coordinates}
  53
  54 Let's for example use a factor 0.75 for $x$, 4
  55 for $y$ and 0.5 for $z$ using the function \texttt{scaleOpoint3d} from the
  56  \textit{jps} library---so a cube will be transformed to a cuboid.
  57 \begin{LTXexample}[pos=t]
  58 \psset{viewpoint=20 60 20 rtp2xyz,lightsrc=viewpoint,Decran=20}
  59 \begin{pspicture}(-5,-3)(6,5)
  60 \psSolid[object=grille,base=-4 4 -4 4,fillcolor=red!50]%
  61 \axesIIID(0,0,0)(4,4,4)%
  62 \psSolid[object=cube,fillcolor=yellow!50,
  63       a=2,ngrid=3](-2,0,1)
  64 \psSolid[object=cube,fillcolor=green!50,
  65       a=2,transform={.75 4 .5 scaleOpoint3d},
  66       ngrid=3](2,0,1)
  67 \psSolid[object=cube,
  68       action=draw,
  69       a=2,ngrid=3](2,0,1)
  70 \end{pspicture}
  71 \end{LTXexample}
  72
  73 \subsection{Transformation associated with the distance to the origin}
  74
  75 Here an example applied to a cube:
  76
  77 \begin{equation*}
  78 \left\lbrace\begin{aligned}
  79 x'&=\big(0.5\sqrt{x^2+y^2+z^2}+1-0.5\sqrt{3}\big)x \\
  80 y'&=\big(0.5\sqrt{x^2+y^2+z^2}+1-0.5\sqrt{3}\big)y \\
  81 z'&=\big(0.5\sqrt{x^2+y^2+z^2}+1-0.5\sqrt{3}\big)z
  82 \end{aligned}\right.
  83 \end{equation*}
  84
  85 \begin{LTXexample}[width=7cm]
  86 \begin{pspicture}(-3,-4)(4,3)
  87 \psset{viewpoint=20 60 20 rtp2xyz,lightsrc=10 15 7,Decran=20}
  88 \pstVerb{
  89 /gro {
  90 4 dict begin
  91    /M defpoint3d
  92    /a .5 def
  93    /b 1 a 3 sqrt mul sub def
  94    /k M norme3d a mul b add def
  95    M k mulv3d
  96 end
  97 } def}%
  98 \psset{linewidth=.02,linecolor=gray}
  99 \psSolid[object=cube,a=3,ngrid=9,
 100       transform=gro]%
 101 \end{pspicture}
 102 \end{LTXexample}
 103 %\newpage
 104
 105 \subsection{Bending and \Index{torsion} of beams}
 106
 107 The solid to the left is a prism of the height 10 cm with 20 floors
 108 (\texttt{\Lkeyword{ngrid}=20 2}). In every floor, an additional angle of rotation---for example 10$^{\mathrm{o}}$ around the $Oz$ axis is---given.
 109  Now that the adjacent floors have a distance of $0.5$~cm, one multiplies $z\times20$.
 110
 111 La flexion est envisag\'{e}e dans le plan $xOz$ sous l'action d'une force perpendiculaire \`{a} la poutre appliqu\'{e}e en son extr\'{e}mit\'{e}.
 112
 113 \begin{LTXexample}[pos=t]
 114 \psset{viewpoint=100 50 20 rtp2xyz,lightsrc=viewpoint,Decran=100,unit=0.65}
 115 \begin{pspicture}(-3,-1)(3.5,11)
 116 \psSolid[object=grille,base=-2 2 -2 2,ngrid=8]%
 117 \psSolid[object=prisme,h=10,ngrid=20 2,
 118         base=0.5 0 0.5 0.5 0 0.5 -0.5 0.5 -0.5 0 -0.5 -0.5 0 -0.5 0.5 -0.5]%
 119 \end{pspicture}
 120 \begin{pspicture}(-3,-1)(3.5,11)
 121 \psSolid[object=grille,base=-2 2 -2 2,ngrid=8]%
 122 \pstVerb{
 123 /torsion {% on tourne de 10 degr\'{e}s suivant l'axe Oz \`{a} chaque niveau
 124 2 dict begin
 125    /M defpoint3d % on r\'{e}cup\`{e}re les coordonn\'{e}es
 126    M /z exch def pop pop
 127     M 0 0 z 20 mul rotateOpoint3d
 128 end} def}%
 129 \psSolid[object=prisme,h=10,ngrid=20 2,
 130         base=0.5 0 0.5 0.5 0 0.5 -0.5 0.5 -0.5 0 -0.5 -0.5 0 -0.5 0.5 -0.5,
 131         transform=torsion]%
 132 \psTransformPoint[RotZ=20](2 0 10)(0,0,0){A}
 133 \psTransformPoint[RotZ=20](2 1 10)(0,0,0){A'}
 134 \psTransformPoint[RotZ=20](-2 0 10)(0,0,0){B}
 135 \psTransformPoint[RotZ=20](-2 -1 10)(0,0,0){B'}
 136 \psline[linecolor=red]{v-v}(A')(A)(B)(B')
 137 \end{pspicture}
 138 \begin{pspicture}(-3.5,-1)(3,11)
 139 \psSolid[object=grille,base=-2 2 -2 2,ngrid=8]%
 140 \pstVerb{%  id\'{e}e de Christophe Poulain
 141 /flexion {% on tourne de 2 degr\'{e}s suivant l'axe Oy \`{a} chaque niveau
 142 2 dict begin
 143    /M defpoint3d % on r\'{e}cup\`{e}re les coordonn\'{e}es
 144    M /z exch def pop pop
 145     M 0 z 2 mul 0 rotateOpoint3d
 146 end} def}%
 147 \axesIIID(0,0,0)(3,3,10)
 148 \psSolid[object=prisme,h=10,ngrid=20 2,
 149         base=0.5 0 0.5 0.5 0 0.5 -0.5 0.5 -0.5 0 -0.5 -0.5 0 -0.5 0.5 -0.5,
 150         transform=flexion]%
 151 \psTransformPoint[RotY=20](0.5 0 10)(0,0,0){A}
 152 \psPoint(3 20 cos mul 20 sin 10 mul add 0.5 add,0, 20 cos 10 mul 20 sin 3 mul sub){A'}
 153 \psdot(A)\psline[linecolor=red]{-v}(A)(A')
 154 \end{pspicture}
 155 \end{LTXexample}
 156
 157 \endinput